Interplay between charge and spin noise in the near-surface theory of decoherence and relaxation of $C_{3v}$ symmetry qutrit spin-1 centers

Decoherence and relaxation of solid-state defect qutrits near a crystal surface, where they are commonly used as quantum sensors, originates from charge and magnetic field noise. A complete theory requires a formalism for decoherence and relaxation that includes all Hamiltonian terms allowed by the defect's point-group symmetry. This formalism, presented here for the $C_{3v}$ symmetry of a spin-1 defect in a diamond, silicon cardide, or similar host, relies on a Lindblad dynamical equation and clarifies the relative contributions of charge and spin noise to relaxation and decoherence, along with their dependence on the defect spin's depth and resonant frequencies. The calculations agree with the experimental measurements of Sangtawesin $\textit{et al.}$, Phys. Rev. X $\textbf{9}$, 031052 (2019) and point to an unexpected importance of charge noise.